The superconducting and magnetic properties of phase-separated AxFe2-ySe2 compounds are known to depend on postgrowth heat treatments and cooling profiles. This paper focuses on the evolution of microstructure on annealing and how this influences the superconducting properties of RbxFe2-ySe2 single crystals. We find that the minority phase in the as-grown crystal has increased unit cell anisotropy (c/a ratio), reduced Rb content, and increased Fe content compared to the main phase. The microstructure is rather complex, with two-phase mesoscopic plate-shaped features aligned along {113} habit planes. The minority phases are strongly faceted on the {113} planes, which we have shown to be driven by minimizing the volume strain energy introduced as a result of the phase transformation. Annealing at 488 K results in coarsening of the mesoscopic plate-shaped features and the formation of a third distinct phase. The subtle differences in structure and chemistry of the minority phase(s) in the crystals are thought to be responsible for changes in the superconducting transition temperature. In addition, scanning photoemission microscopy has clearly shown that the electronic structure of the minority phase has a higher occupied density of states of the low binding energy Fe3d orbitals, which is characteristic of crystals that exhibit superconductivity. This demonstrates a clear correlation between the Fe-vacancy-free phase with high c/a ratio and the electronic structure characteristics of the superconducting phase. © 2014 American Physical Society.